Let A be a real continuous nxn matrix on an interval T3 and let the n-vector x be a solution of the differential equation x = Ax on r. If [oc,g]er, g is called a conjugate point of a if the equation has a nontrivial solution vector x = (x1,...,xn) such that x1(a) = ... = xk(a) = x k+1(P) = ... = x (0) = 0 for some ke[l,n-l]. It is shown that the absence on (t..,t ) of a point conjugate to t, wi...

Let G be a finite set of points in the plane. A line M is a (k, k)-line, if M is determined by G, and there are at least k points of G in each of the two open half-planes bounded by M . Let f(k, k) denote the maximum size of a set G in the plane, which is not contained in a line and does not determine a (k, k)-line. In this paper we improve previous results of Jaakov Kupitz (f(k, k) ≤ 3k), Noga...

The relationships between the values taken by ordinary characters of symmetric groups are exploited to prove two theorems in the modular representation theory of the symmetric group. 1. The decomposition matrices of symmetric groups in odd characteristic have distinct rows. In characteristic 2 the rows of a decomposition matrix labelled by the different partitions λ and μ are equal if and only ...